On sequence space representation and extension of vector-valued functions

We give a unified approach to handle the problem of extending functions with values in a locally convex Hausdorff space E over a field K, which have weak extensions in a space .T(O, K) of scalar-valued functions on a set O, to functions in a vector-valued counterpart .T(O, E) of .T(O, K). The results obtained are based upon a representation of vector-valued functions as linear continuous operators and extend results of Bonet, Frerick, Gramsch and Jorda. In particular, we apply them to obtain a sequence space representation of .T(O, E) from a known representation of .T(O, K).